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Full text of "Treatise On Analysis Vol-Ii"

20 BOUNDED MEASURES 217 Let 3~i denote the weak topology on Mi(X) corresponding to the vector space Ei (i ¥= 4), and ^~4. v the weak topology on Mi(X, v) (or on L£(X, v)) corresponding to E4,v(cf. (12.15)). The topology &~t on Mi(X) is coarser than &~j if / < j. The topology induced by .^"3 on MA(X, v) is coarser than ^"4fV, which in turn is coarser than the topology induced by ^5 - (Consider the lower semicontinuous regularization of a function which is continuous almost everywhere with respect to v (Section 12.7, Problem 8).) (a) Let (p,n) be a sequence of bounded real measures on a locally compact but not compact space X. Give an example in which (/zn) tends to 0 vaguely (i.e., for the topo- logy fi) but does not converge for the topology &"2. A sequence (ju,n) which vaguely converges to 0 also converges to 0 with respect to 3~2 if and only if the sequence of norms (||/xn||) is bounded (use the Banach-Steinhaus theorem). (b) Give an example of a sequence (jiO which converges to 0 for the topology #"a but not for ^"3 - A sequence (/un) of bounded real measures which converges vaguely to a measure /z also converges to /* for the topology ^"3 if and only if, for each e > 0, there exists a compact subset K of X such that |/xn|(X — K)^ s for all n. (To show that the condition is necessary, argue by contradiction, by using the method of Section 13.14, Problem 1.) (c) Give an example of a sequence (fO in Mi(X, v) which converges to 0 for the topology ^"3 but does not converge to 0 for the topology ^~4, v (take X = [0,1 ]). A sequence (jitn) of measures belonging to M£(X, v) which converges to 0 for the topol- ogy &~3 also converges to 0 with respect to ,?"4, v if and only if it satisfies the following condition: (C4, v) For each v-negligible compact subset K of X and each s > 0, there exists an open neighborhood U of K such that |ftn|(U) •<* e for all n. (To prove that the condition is sufficient, reduce to the case where X is compact and apply (€4, v) to the set Ke of points x e X at which the oscillation of an almost everywhere continuous function is ^.s. To prove that the condition is necessary, argue by contradiction as in (b) above.) (d) Show that a sequence (/*„) in M^(X) which is a Cauchy sequence for one of the topologies ^"i, ^2, ^"3 is convergent for this topology. (For ^"3, use (b) and argue by contradiction: form a sequence of measures which tends to 0 with respect to ^"3 without satisfying the condition given in (b).) (e) A sequence (ju,n) of measures belonging to Mi(X, v) is a Cauchy sequence with respect to the topology ^~4, v if and only if it satisfies the condition (C4, v) (argue by contradiction as in (d)); the sequence (fjLn) then converges with respect to «^"4, v to a measure belonging to Mi(X, v). (f) Let (jUH) be a sequence of measures belonging to M^(X, v). For each subset A of X which is either finite or else open and v-quadrable (Section 13.9, Problem 7), suppose that the sequence (ftn(A)) has a finite limit; show that (fjLn) is then a Cauchy sequence for the topology ^"4,v. (Use Problem 1 of Section 13.14, and argue by contradiction.) Show that the hypothesis relative to finite sets A cannot be omitted. (g) Let (jLtn) be a sequence of measures belonging to M^(X, v) which converges in the topology ^"3 to a measure ju, belonging to M^(X, v). If also lim ||ju,n|| = ||/A||, then H-+00 the sequence (^u,,) converges to fi in the topology «^~4, v (use (e)). Give an example of a sequence (fjin) of positive measures belonging to M£(X, v), where the space X is compact, such that (jitw) converges vaguely to a measure which does not belong to M£(X, v). The notation is the same as in Problem 1. Let (ftn) be a sequence of measures belonging to M^(X). Then there exists a positive measure v on X such that JLC« e M^(X, v) for all n.